Invariant Measures for Soliton Systems Generated by Mealy Automata

Abstract

We study invariant measures for soliton systems described by Mealy automata. Motivated by recently introduced soliton models associated with 2-letter, 3-state Mealy automata, we formulate the time evolution induced by Mealy automata on bi-infinite configuration spaces. We provide sufficient conditions for the invariance of Bernoulli product measures and derive a criterion for the invariance of two-sided space-homogeneous Markov distributions. We then apply these general results to three soliton models, which can be interpreted as variants of the box-ball system (BBS). For two of these models, BBS-S(2) and BBS-V(2), we prove that Bernoulli product measures are invariant. For the remaining model, BBS-C(2), we establish a more general result: the invariance of two-sided space-homogeneous Markov distributions, which include Bernoulli product measures as a special case. Furthermore, for all three models, we compute the phase shift associated with the interaction of two solitons, as well as the velocity of an isolated soliton. Although the latter has already been studied previously, both quantities constitute fundamental characteristics for understanding the generalized hydrodynamics of these systems. These results provide a foundation for the study of invariant measures, generalized Gibbs ensembles, and generalized hydrodynamic behavior in Mealy-automaton soliton systems.